User blog:ArtismScrub/Hyper-Alpha Notation Part II: Arrows to Alphas (STILL WIP)
See Hyper-Alpha Notation . Last time, I had defined up to A↑↑A, which reached strength of ε0. Now, I've found a simple method to generalize this towards pentation, hexation, etc. (I actually came up with this some time before I even made the original blog post, but I hadn't bothered to test it to its limits.) Consider the properties: An = AAAAAA........AAAAAA with n "A"s A★A = (A★)A UPGRADE: A ↑a+1 b = A ↑a A ↑a A ↑a ... ↑a A ↑a A ↑a A with b "A"s A ↑n ★A = (A ↑n ★) ↑n A This simple change will cover us for any number of up-arrows. Of course, like last time, I'll give some recursive progression. A↑↑A ≈ ε0 (this was where we left off last time) (A↑↑A)A ≈ ε0+1 (A↑↑A)An ≈ ε0+n (A↑↑A)AA ≈ ε0+ω (A↑↑A)(A↑↑n) ≈ ε0+(ω↑↑(n-1)) (A↑↑A)(A↑↑A) ≈ ε02 (A↑↑A)n ≈ ε0n (A↑↑A)A ≈ ε0ω = ωε0+1 (A↑↑A)AA ≈ ε0(ω2) (A↑↑A)AA ≈ ε0(ωω) (A↑↑A)(A↑↑n) ≈ ε0(ω↑↑n) (A↑↑A)(A↑↑A) ≈ ε02 (A↑↑A)(A↑↑A)A ≈ ε02ω (A↑↑A)(A↑↑A)(A↑↑A) ≈ ε03 (A↑↑A)(A↑↑A)A ≈ ε0ω (A↑↑A)(A↑↑A)AA ≈ ε0ωω (A↑↑A)(A↑↑A)(A↑↑n) ≈ ε0(ω↑↑n) (A↑↑A)(A↑↑A)(A↑↑A) ≈ ε0ε0 (A↑↑A)(A↑↑A)(A↑↑A)(A↑↑A) ≈ ε0ε0ε0 (A↑↑A)↑↑n ≈ ε0↑↑(n-1) (A↑↑A)↑↑A = A↑↑AA ≈ ε1 (A↑↑AA)A ≈ ε1ω (A↑↑AA)(A↑↑A) ≈ ε1×ε0 (A↑↑AA)(A↑↑AA) ≈ ε12 (A↑↑AA)(A↑↑AA)A ≈ ε1ω (A↑↑AA)(A↑↑AA)(A↑↑A) ≈ ε1ε0 (A↑↑AA)(A↑↑AA)(A↑↑AA) ≈ ε1ε1 (A↑↑AA)(A↑↑AA)(A↑↑AA)(A↑↑AA) ≈ ε1ε1ε1 A↑↑AAA ≈ ε2 (A↑↑AAA)(A↑↑AAA) ≈ ε22 (A↑↑AAA)(A↑↑AAA)(A↑↑AAA) ≈ ε2ε2 A↑↑AAAA ≈ ε3 A↑↑An ≈ εn-1 A↑↑AA ≈ εω A↑↑(AAA) ≈ εω+1 A↑↑(AAAA) ≈ εω2 A↑↑(AAA) ≈ εω2 A↑↑(AAA) ≈ εωω A↑↑(A↑↑n) ≈ εω↑↑n A↑↑(A↑↑A) ≈ εε0 A↑↑(A↑↑(A↑↑A)) ≈ εεε0 A↑↑↑A ≈ ζ0 (A↑↑↑A)A ≈ ζ0ω (A↑↑↑A)(A↑↑A) ≈ ζ0×ε0 (A↑↑↑A)(A↑↑↑A) ≈ ζ02 (A↑↑↑A)(A↑↑↑A)A ≈ ζ0ω (A↑↑↑A)(A↑↑↑A)(A↑↑A) ≈ ζ0ε0 (A↑↑↑A)(A↑↑↑A)(A↑↑↑A) ≈ ζ0ζ0 (A↑↑↑A)(A↑↑↑A)(A↑↑↑A)(A↑↑↑A) ≈ ζ0ζ0ζ0 (A↑↑↑A)↑↑A ≈ εζ0+1 (A↑↑↑A)↑↑AA ≈ εζ0+2 (A↑↑↑A)↑↑AA ≈ εζ0+ω (A↑↑↑A)↑↑(A↑↑A) ≈ εζ0+ε0 (A↑↑↑A)↑↑(A↑↑AA) ≈ εζ0+ε1 (A↑↑↑A)↑↑(A↑↑AA) ≈ εζ0+εω (A↑↑↑A)↑↑(A↑↑(A↑↑A)) ≈ εζ0+εε0 (A↑↑↑A)↑↑(A↑↑↑A) ≈ εζ02 (A↑↑↑A)↑↑(A↑↑↑A)A ≈ εζ0ω (A↑↑↑A)↑↑(A↑↑↑A)(A↑↑↑A) ≈ εζ02 (A↑↑↑A)↑↑(A↑↑↑A)(A↑↑↑A)(A↑↑↑A) ≈ εζ0ζ0 (A↑↑↑A)↑↑(A↑↑↑A)↑↑A ≈ εεζ0+1 (A↑↑↑A)↑↑(A↑↑↑A)↑↑(A↑↑↑A) ≈ εεζ02 A↑↑↑AA ≈ ζ1 (A↑↑↑AA)↑↑A ≈ εζ1+1 (A↑↑↑AA)↑↑(A↑↑A) ≈ εζ1+ε0 (A↑↑↑AA)↑↑(A↑↑↑A) ≈ εζ1+ζ0 (A↑↑↑AA)↑↑(A↑↑↑AA) ≈ εζ12 (A↑↑↑AA)↑↑(A↑↑↑AA)↑↑(A↑↑↑AA) ≈ εεζ12 A↑↑↑AAA ≈ ζ2 A↑↑↑AA ≈ ζω A↑↑↑(A↑↑A) ≈ ζε0 A↑↑↑(A↑↑↑A) ≈ ζζ0 A↑↑↑(A↑↑↑(A↑↑↑A)) ≈ ζζζ0 A↑↑↑↑A ≈ η0 (A↑↑↑↑A)A ≈ η0ω (A↑↑↑↑A)↑↑A ≈ εη0+1 (A↑↑↑↑A)↑↑↑A ≈ ζη0+1 A↑↑↑↑AA ≈ η1 A↑↑↑↑AA ≈ ηω A↑↑↑↑(A↑↑A) ≈ ηε0 A↑↑↑↑(A↑↑↑A) ≈ ηζ0 A↑↑↑↑(A↑↑↑↑A) ≈ ηη0 A↑↑↑↑(A↑↑↑↑(A↑↑↑↑A)) ≈ ηηη0 A↑↑↑↑↑A ≈ φ(4,0) A↑↑↑↑↑↑A ≈ φ(5,0) A ↑n A ≈ φ(n-1,0) A ↑A A ≈ φ(ω,0) A ↑AA A and beyond: How to work with these? Only God knows--but they must form some massive alpha strings--and utterly unspeakable numbers when solved. Nah, just kidding, I haven't generalized that far yet! Just the previous Thursday, I was shocked to reach ε0. I hadn't expected to reach such recursive heights, and now I've made it FAR further than that. This is certainly a milestone. How many people have even managed φ(ω,0) by themselves? Probably a minority out of all. Again, I do hope to generalize this further for higher A ↑★ A, and of course, iterate ★ and form a fully functional legiattic notation, reaching ψ(Ωω) and beyond. This will either be a simple modification or a distant dream. Category:Blog posts